Hyperbolic Tessellations Associated to Bianchi Groups

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Abstract

Let F/ℚ be a number field. The space of positive definite binary Hermitian forms over F form an open cone in a real vector space. There is a natural decomposition of this cone into subcones. In the case of an imaginary quadratic field these subcones descend to hyperbolic space to give rise to tessellations of 3-dimensional hyperbolic space by ideal polytopes. We compute the structure of these polytopes for a range of imaginary quadratic fields.